Do
I contradict myself? Very well then, I contradict myself. I am large, I
contain multitudes.

-Walt Whitman

The most widely accepted formal basis for arithmetic is
called Peano's Axioms (PA). Giuseppe Peano based his system on a specific
natural number, 1, and a successor function such that to each natural number
x there corresponds a successor x' (also denoted as x+1). He then formulated
the properties of the set of natural numbers in five axioms:

(1) 1 is a
natural number.

(2) If x is
a natural number then x' is a natural number.

(3) If x is
a natural number then x' is not 1.

(4) If x' =
y' then x = y.

(5) If S is
a set of natural numbers including 1, and if for every x in S the successor
x' is also in S, then every natural number is in S.

These axioms (together with numerous tacit rules of
reasoning and implication, etc) constitute a formal basis for the subject of
arithmetic, and all formal “proofs” ultimately are derived from them. The
first four, at least, appear to be “clear and distinct” notions, and even the
fifth would be regarded by most people as fairly unobjectionable.
Nevertheless, the question sometimes arises (especially in relation to very
complicated and lengthy proofs) whether theorems based on these axioms (and
tacit rules of implication) are perfectly indubitable. According to Goedel’s
theorem, it is impossible to formally prove the consistency of arithmetic, which
is to say, we have no rigorous proof that the basic axioms of arithmetic do
not lead to a contradiction at some point. For example, if we assume some
proposition (perhaps the negation of a conjecture we wish to prove), and
then, via some long and complicated chain of reasoning, we arrive at a
contradiction, how do we know that this contradiction is essentially a
consequence of the assumed proposition? Could it not be that we have just
exposed a contradiction inherent in our formal system of arithmetic itself?
In other words, if arithmetic itself is inconsistent, then proof by contradiction
loses its persuasiveness.

On one level, this kind of objection can easily be
vitiated by simply prefacing every theorem with the words "If our
formalization of arithmetic is consistent, then...". Indeed, for short
simple proofs by contradiction we can strengthen the theorem by reducing this
antecedent condition to something like "If arithmetic is consistent over
this small set of operations, then...". We can be confident that the contradiction
really is directly related to our special assumption, because it's highly
implausible that our formalization of arithmetic could exhibit a
contradiction over a very short chain of implication. However, with long
proofs of great subtlety, extending over multiple papers by multiple authors,
and involving the interaction of many different branches and facets of
mathematics, how would we really distinguish between a subtle contradiction
resulting from one specific false assumption vs. a subtle contradiction
inherent in the fabric of arithmetic itself?

Despite Goedel’s theorem, the statement that we cannot
absolutely prove the axioms of arithmetic is sometimes challenged on the
grounds that we can prove the consistency of PA, provided we are willing to
accept the consistency of some more encompassing formal system such as the
Zermelo-Frankel (ZF) axioms, perhaps augmented with the continuum hypothesis
(ZFC). But this is a questionable position. Let's say a proof of the
consistency of system X is "incomplete" if it's carried out within
a system Y whose consistency has not been completely proven. Theorem:
Every proof of the consistency of arithmetic is incomplete. In view of this,
it isn't clear how "working in ZFC" resolves the issue. There is
no complete and absolute proof of the consistency of arithmetic, so every
arithmetical proof is subject to doubt. (By focusing on arithmetic I don't
mean to imply that other branches of mathematics are exempt from doubt.
Hermann Weyl, commenting on Gödel’s work, said that "God exists because
mathematics is undoubtedly consistent, and the devil exists because we cannot
prove the consistency".)

As Morris Kline said in his book Mathematics and the
Loss of Certainty, “Gödel’s result on consistency says that we cannot
prove consistency in any approach to mathematics by safe logical
principles”, meaning first-order logic and finitary proof theory, which had
been shown in Russell's "Principia Mathematica" to be sufficient as
the basis for much of mathematics. Similarly, in John Stillwell's book Mathematics
and its History we find "If S is any system that includes PA, then
Con(S) [the consistency of S] cannot be proved in S, if S is
consistent." On the other hand, some would suggest that the
contemplation of inconsistency in our formal arithmetic is tantamount to a renunciation
of reason itself, i.e., if our concept of natural numbers is inconsistent
then we must be incapable of rational thought, and any further considerations
are pointless. This attitude is reminiscent of the apprehensions
mathematicians once felt regarding "completed infinities".
"We recoil in horror", as Hermite said, believing that the
introduction of actual infinities could lead only to nonsense and sophistry.
Of course, it turned out that we are quite capable of reasoning in the
presence of infinities. Similarly, I believe reason can survive even the
presence of contradiction in our formal systems.

Admittedly this belief is based on a somewhat unorthodox
view of formal systems, according to which such systems should be seen not as
unordered ("random access") sets of syllogisms, but as structured
spaces, with each layer of implicated objects representing a region, and the
implications representing connections between different regions. The space
may even possess a kind of metric, although "distances" are not
necessarily commutative. For example, the implicative distance from an
integer to its prime factors is greater than the implicative distance from those
primes to their product. According to this view a formal system does not
degenerate into complete nonsense simply because at some point it contains a contradiction.
A system may be "locally" consistent even if it is not globally
consistent. To give a crude example, suppose we augment our normal axioms
and definitions of arithmetic with the statement that a positive integer n is
prime if and only if 2n-
2 is divisible by n. This axiom conflicts with our existing definition of a
prime, but the first occurrence of a conflict is 341. Thus, over a limited
range of natural numbers the axiom system possesses "local
consistency".

Suppose we then substitute a stronger axiom by saying n is
a prime iff f(rn) = 0 (mod n) where r is any root of f(x) = x5- x3- 2x2 + 1. With this system we
might go quite some time without encountering a contradiction. When we
finally do bump into a contradiction (e.g., 2258745004684033) we could simply
substitute an even stronger axiom. In fact, we can easily specify an axiom
of this kind for which the smallest actual exception is far beyond anyone's
(present) ability to find, and for which we have no theoretical proof that
any exception even exists. Thus, there is no direct proof of inconsistency.
We might then, with enough imagination, develop a plausible (e.g., as
plausible as Banach-Tarski) non-finitistic system within which I can actually
prove that our arithmetic is consistent. In fact, it might actually be
consistent… but we would have no more justification to claim absolute certainty
than with our present arithmetic.

As to the basic premise that we have no absolute proof of
the consistency of arithmetic, here are a few other people's thoughts on the
subject:

A
meta-mathematical proof of the consistency of arithmetic is not excluded
by...Goedel's analysis. In point of fact, meta-mathematical proofs of the
consistency of arithmetic have been constructed, notably by Gerhard Gentzen,
a member of the Hilbert school, in 1936. But such proofs are in a sense
pointless if, as can be demonstrated, they employ rules of inference whose own
consistency is as much open to doubt as is the formal consistency of
arithmetic itself. Thus, Gentzen used the so-called "principle of
transfinite mathematical induction" in his proof. But the principle in
effect stipulates that a formula is derivable from an infinite class
of premises. Its use therefore requires the employment of nonfinitistic meta
- mathematical notions, and so raises once more the question which Hilbert's
original program was intended to resolve.

-Ernest Nagel and James Newman

Gödel
showed that...if anyone finds a proof that arithmetic is consistent, then it
isn't!

-Ian Stewart

...Hence
one cannot, using the usual methods, be certain that the axioms of arithmetic
will not lead to contradictions.

-Carl Boyer

An
absolute consistency proof is one that does not assume the consistency of
some other system...what Gödel did was show that there must be
"undecidable" statements within any [formal system]... and
that consistency is one of those undecidable propositions. In other words,
the consistency of an all-embracing formal system can neither be proved nor
disproved within the formal system.

-Edna
Kramer

Gentzen's
discovery is that the Goedel obstacle to proving the consistency of number
theory can be overcome by using transfinite induction up to a sufficiently
great ordinal... The original proposals of the formalists to make classical
mathematics secure by a consistency proof did not contemplate that such a
method as transfinite induction up to e0 would have to be used. To what extent the Gentzen
proof can be accepted as securing classical number theory in the sense of
that problem formulation is in the present state of affairs a matter for individual
judgment...

-Kleene, "Introduction to
Metamathematics"

Some mathematicians assert that there is a consistency
proof of PA, and it is quite elementary, using standard mathematical
techniques (ie, ZF). It consists of exhibiting a model. However, we speak of
"exhibiting a model" we are referring to a relative consistency
proof, not an absolute consistency proof. Examples of relative consistency
theorems are

If Euclidean geometry is
consistent then non-Euclidean geometry is consistent.

If ZF is consistent then
ZFC is consistent.

Relative consistency proofs assert nothing about the
absolute consistency of any system, they merely relate the consistency of one
system to that of another. Here's what the Encyclopedic Dictionary of Mathematics
(2nd Ed) says on the subject:

Hilbert
proved the consistency of Euclidean geometry by assuming the consistency of
the theory of real numbers. This is an example of a relative consistency
proof, which reduces the consistency proof of one system to that of another.
Such a proof can be meaningful only when the latter system can somehow be
regarded as based on sounder ground than the former. To carry out the
consistency of logic proper and set theory, one must reduce it to that of
another system with sounder ground. For this purpose, Hilbert initiated
metamathematics and the finitary standpoint...Let S be any consistent formal
system containing the theory of natural numbers. Then it is impossible to
prove the consistency of S by utilizing only arguments that can be formalized
in S.... In [these] consistency proofs of pure number theory..., transfinite
induction up to the first e-number e0 is used, but all the other reasoning used in these
proofs can be presented in pure number theory. This shows that the legitimacy
of transfinite induction up to e0 cannot be proved in this latter theory.

All known consistency proofs of arithmetic rely on
something like transfinite induction (or possibly primitive recursive
functionals of finite type), the consistency of which is no more self-evident
than that of arithmetic itself.

Oddly enough, some people (even some mathematicians) are
under the impression that Goedel’s results apply only to very limited formal
systems. One mathematician wrote to me that “there is no proof in first-order
logic that arithmetic is consistent, but that has more to do with the
limitations of first-order logic than anything else, and there are other more
general types of logic in which proofs of the consistency of arithmetic are
available.” Of course, contrary to this individual’s claim, Goedel's results actually
apply to any formal system that is sufficiently complex to encompass
and be modeled by arithmetic. Granted, if we postulate a system that cannot
be modeled (encoded) by arithmetic then other things are possible, but the
consistency of such a system would be at least as doubtful as the consistency
of the system we were trying to prove. For example, Gentzen's proof of the
consistency of PA uses transfinite induction, but surely it is pointless to
try to resolve doubts about arithmetic by working with transfinite induction,
since the latter is even more dubious.

The inability of even many mathematicians to absorb the
actual content and significance of Goedel’s theorems is interesting in
itself, as are the various misconstruals of those theorems, which tend to
reflect what the person thinks must be true. For example, we can see how
congenial is the idea that Goedel’s results apply only to a limited class of
formal systems. The fact that mathematicians at universities actually believe
this is rather remarkable. It seems to be a case of sophomoric backlash, in
reaction to what can often seem like sensationalistic popular accounts of
Goedel’s theorems. Apparently it becomes a point of pride among math graduate
students to “see through the hype”, and condescendingly advise the less
well-educated as to the vacuity of Goedel’s theorems. (This would be fine if
Goedel’s theorems actually were vacuous, but since they aren’t, it isn’t.)
Another apparent source of misunderstanding is the sheer inability to believe
that any rational person could doubt the consistency of, say, arithmetic.

Imagine the reaction of the typical mathematician to the
even more radical suggestion that every (sufficiently complex) formal system
contains a contradiction at some point. When I mentioned this to an email
correspondent, he expressed utter incredulity, saying "You can't
possibly believe that simple arithmetic could contain an inconsistency! How
would you balance your check book?" This is an interesting question. I
actually balance my checkbook using a formal system called Quicken. Do I have
a formal proof of the absolute consistency and correctness of Quicken? No. Is
it conceivable that Quicken might contain an imperfection that could lead, in
some circumstances, to an inconsistency? Certainly. But for many
mathematicians this situation must be a real paradox, so it’s worth examining
in some detail.

Suppose I balance my checkbook with a program called Kash
(so as not to sully the good name of Quicken), and suppose this program implements
arithmetic perfectly - with one exception. The result of subtracting
5.555555 from 7.777777 is 2.222223. Now if B is my true balance then I
should have the formal theorems

for every value of q. Thus, in the formal system
of Kash I can prove that 1 = 2 = 3 = 4.23 = 89.23 = anything. Clearly the
Kash system has a consistency problem, because I can compute my balance to be
anything I want just by manipulating the error produced by that one
particular operation. ("So oft it chances in particular men...")
But here is the fact that must seem paradoxical to many mathematicians:
thousands of people have used Kash for years, and not a single error has appeared
in the results. How can this be, given that Kash is formally inconsistent?

The answer is that although Kash is globally inconsistent,
it possesses a high degree of local consistency. Traveling from any given
premise (such as 5+7) directly to the evaluation (e.g., 12), we are
very unlikely to encounter the inconsistency. Of course, in a perfectly
consistent system we could take any of the infinitely many paths from
5+7 to the evaluation and we would always arrive at the same result, which is
clearly not true within Kash (in which we could, by round-about formal
manipulations, evaluate 5+7 to be -3511.1093, or any other value we wanted).
Nevertheless, for almost all paths leading from a given premise to its
evaluation, the result is the same.

Now consider our formal system of arithmetic. Many people
seem agog at the suggestion that our formalization of arithmetic might
possibly be inconsistent at some point. Clearly our arithmetic must possess a
very high degree of local consistency, because otherwise we would have
observed anomalies long before now. However, are we really justified in
asserting that every one of the infinitely many paths from every
premise to its evaluation gives the same result? As with the system Kash,
this question can't be answered simply by observing that our checkbooks
usually seem to balance.

Moreover, the question cannot even be answered within any
formal system that can be modeled by the natural numbers. It is evidently necessary
to assume the validity of something like transfinite induction to prove the
consistency of arithmetic. But how sure are we that a formal system that
includes transfinite induction is totally consistent? (If, under the
assumption of transfinite induction, we had found that arithmetic was not
consistent, would we have abandoned arithmetic... or transfinite induction?)
The only way we know how to prove this is by assuming still less
self-evidently consistent procedures, and so on.

The points I'm trying to make are

(1) We have
no meaningful proof of the consistency of arithmetic.

(2) If
arithmetic is inconsistent, it does not follow that our
checkbooks must all be out of balance. It is entirely possible that we could
adjust our formalization of arithmetic to patch up the inconsistency, and almost
all elementary results would remain unchanged.

(3) However,
highly elaborate and lengthy chains of deduction in the far reaches of
advanced number theory might need to be re-evaluated in the light of our
patched-up formalization.

Of course, the consistency or inconsistency of arithmetic can
only be appraised in the context of a completely formalized system, but the
very act of formalizing is problematic, because it invariably presupposes prior
knowledge on the part of the reader. Thus it can never be completely clear
that our formalization corresponds perfectly with what we call `arithmetic'. Our
efforts to project (exert) our formalizations past any undefined prior
knowledge tend to lead to the possibility of contradictions and
inconsistencies. But when such an inconsistency comes to light, we don’t say
the ideal of “arithmetic” is faulty. Rather we say "Ooppss, our
formalization didn't quite correspond to true ideal arithmetic. Now,
here's the final ultimate and absolutely true formalization... (I know we
said this last time, but this time our formalization really is
perfect.)" As a matter of fact, this very thing has occurred
historically. In this sense we are tacitly positing the existence of a
Platonic ideal "ARITHMETIC" that is eternal, perfect, and true,
while acknowledging that any given formalization of this Platonic ideal may
be flawed. The problem is that our formal proofs are based on a specific formalization
of arithmetic, not on the ideal Platonic ARITHMETIC, so we are not justified
in transferring our sublime confidence in the Platonic ideal onto our formal
proofs.

Claims that arithmetic is indubitable, while acknowledging
that our formalization may not be perfect, are essentially equivalent to
saying that we are always right, because even if we are found to have said
something wrong, that's not what we meant. Any given theorem can be regarded
as a theorem about the ideal of ARITHMETIC, prior to any particular formalization,
but then the first step in attempting to prove it is to select a formal
system within which to work. Of course, it's trivial to devise a formal system
labeled "arithmetic" and then prove X within that system. For
example, take PA+X. But the question is whether that system really
represents ARITHMETIC, one requirement of which is consistency.

We don't know what, if any, parts of our present
mathematics would be rendered uninteresting by the discovery of an
inconsistency in our present formalization of arithmetic, because it would
depend on the nature of the inconsistency and the steps taken to resolve it.
Once the patched-up formalization was in place, we would re-evaluate all of
our mathematics to see which, if any, proofs no longer work in the new
improved "arithmetic". One would expect that almost all present
theorems would survive. The theorems most likely to be in jeopardy would be
the most elaborate, far-reaching, and "deep" results, because their
proofs tax the resources of our present system the most.

Some mathematicians respond to the assertion that we have
no meaningful proof of the consistency of arithmetic by claiming that “the
usual ZFC proof is quite meaningful." But this seems to hinge on
different understandings of the meaning of “meaningful”. Consider the two
well known theorems

(1) con(ZF) implies con(ZFC)

(2) con(ZF) implies con(PA)

From a foundational standpoint, these two theorems act in
opposite directions. In case (1), if the result had been {con(ZF) implies NOTcon(ZFC)}
then it would have undermined our confidence in the "C" part of
ZFC. However, if the result of case (2) had been {con(ZF) implies
NOTcon(PA)}, it would presumably would have undermined our confidence in ZF,
not in PA (because the principles of PA are considered to be more
self-evidently consistent that those of ZF). The only kind of proof that
would enhance our confidence in PA would be of the form

con(X) implies con(PA)

where X is a system whose consistency is MORE self-evident
than that of PA. (This is the key point.) For example, Hilbert hoped that
with X = 1st Order Logic it would be possible to prove this theorem, thereby enhancing
our confidence in the consistency of PA. That would have been a meaningful
proof of the consistency of PA. However, it's now known that such a proof is
impossible (unless you believe in the existence of a formal system that is
more self-evidently consistent than PA but that cannot be modeled within the
system of natural numbers).

Others argue that it may be possible to regard the theory
of primitive recursive functionals as more evidently consistent than PA. It's
well known that both transfinite induction and the theory of primitive
recursive functionals cannot be modeled within the system of natural numbers,
but we do not need to claim that it would be impossible to regard such
principles as more evidently consistent than PA. We simply observe that no
one does – and for good reason. Each represents a non-finitistic extension
of formal principles, which is precisely the source of uncertainty in the
consistency of PA. Again, there is a little thought experiment that sometimes
helps people sort out their own hierarchy of faith: If, assuming the
consistency of the theory of primitive recursive functionals, one could prove
that PA is NOT consistent, would we be more inclined to abandon PA or the
theory of primitive recursive functionals?

Some mathematicians assert that doubts about whether PA is
consistent, and whether it can be proven to be consistent, are trivial and
pointless, partly because this places all of mathematics in doubt. However, as
to the triviality, much of the most interesting and profound mathematics of
this century has been concerned with just such doubts. As to the number of
proofs that are cast into doubt by the possibility of inconsistency in PA,
the "perfect consistency or total gibberish" approach to formal
systems evidently favored by many mathematicians is not really justified.

It just so happened that Russell and Whitehead's FOL was a
convenient finitistic vehicle to use as an example, although subsequent
developments showed that this maps to computability, from which the idea of a
universal Turing machine yields a large segment (if not all) of what can be
called cognition. Of course, people sometimes raise the possibility of a
finitistic system that cannot be modeled within the theory of natural numbers
but, as Ernst Nagel remarked, "no one today appears to have a clear idea
of what a finitistic proof would be like that is NOT capable of formulation
within arithmetic".

PA can be modeled within ZF. It follows that con(ZF)
implies con(PA). This was simply presented as an illustration of how the
formal meaning of a theorem of this form depends on our "a priori"
perceptions of the relative soundness of the two systems.

Some mathematicians have alluded to the "usual"
proof of con(PA) but have not specified the formal system within which this
"usual" proof resides. Since there are infinitely many
possibilities, it's not possible to guess which specific one they have in mind.
In general terms, if there exists a proof of con(PA) within a formal system
X, then we have the meta-theorem { con(X) implies con(PA) }, so we can
replace X with whatever formal system we consider to be the "usual"
one for proofs of PA. The meaningfulness of such a theorem depends on our
perception of the relative soundness of the systems X and PA. I assert that
no system X within which con(PA) can be proved is evidently more consistent
that con(PA) itself.

The
above argument [Goedel's 2nd Thm] can be transferred to other systems where
there is a substitute for the natural numbers and where R-decidable relations
and R-computable functions are representable. In particular, it applies to
systems of axioms for set theory such as ZFC... Since contemporary
mathematics can be based on the ZFC axioms, and since...the consistency of
ZFC cannot be proved using only means available within ZFC, we can formulate
[this theorem] as follows: If mathematics is consistent, we cannot prove its
consistency by mathematical means.

I wanted to include this quote because every time I say
there is no meaningful formal proof of the consistency of arithmetic (PA), someone
always says "Oh yes there is, just work in ZFC, or ZF, or PA + transfinite
induction, or PA + primitive recursive functionals, or PA + con(PA), or just
use the "usual" proof, or (as one mathematician advised me) just stop
and think about it!" But none of these proposed
"proofs" really adds anything to the indubitability of con(PA). Of
course, it's perfectly acceptable to say "I'm simply not interested in
the foundational issues of mathematics or in the meaning of consistency for
formal systems". However, disinterest should not be presented as a
substitute for proof.

In response to the rhetorical question “Are we really justified
in asserting that every one of the infinitely many paths from every
premise to its evaluation gives the same result?”, some mathematicians will
say that if we allow associativity of modus ponens, then there is essentially
only one proof of any classical formula. However, this misses a crucial
point. If we define the "essential path" between any two points in
space as a function only of the beginning and ending points, then there is essentially
only one path from New York to Los Angeles. This definition of a
"path", like the corresponding definition of a "proof" is
highly reductionist. We are certainly free to adopt that definition if we
wish, but in so doing we ignore whole dimensions of non-trivial structure, as
well as making it impossible to reason meaningfully about "imperfectly consistent"
systems which, for all we know, may include every formal system that exists. The
unmeasured application of the modus ponens (rule of detachment) is precisely
what misleads people into thinking that a formal system must be either
absolutely consistent or total gibberish. Then, when they consider systems
such as naive set theory, PA, or ZFC, which are clearly not total gibberish,
they conclude that they must be absolutely consistent.

Some mathematicians claim that PA is a collection of
simple statements, all of which are manifestly true about the positive
integers. However, although the explicit axioms look simple, they are
meaningful only in the context of a vast mathematical apparatus that includes
conventions regarding names of variables, rules about substitutions, and more
generally rules of inference and implication. For example, if we propose a
sequence of variable names x, x', x'' and so on, we need to know that all
these names are different. Of course, most of us would accept this as
intuitively obvious, but if the formal system is to be absolutely watertight,
we must prove that they are different. Thus we end up needing to apply the
Peano axioms to the language used to talk about them.

As Jan Nienhuys observed, one of the basic subtleties
related to Peano’s axioms is that they implicitly assert that any set of
positive integers has a smallest element. In view of Ramsey numbers, it’s
clear that even in simple cases this is more a metaphysical assertion than a matter
of "manifestly true". How can we have confidence that any set of
positive integers, no matter how unwieldy its definition, has a smallest
element? (The unwieldiness of definitions is made possible by having an
unlimited amount of positive integers and variable names at our disposal.)The Peano axioms plus a formal system in which they are embedded must
be free of contradictions, i.e. for no statement X is both X and not-X
provable. (Also, note that we allow variables such as X to denote statements,
so the formal system should have watertight descriptions - no hand waving or illustration
by example allowed - of how to go about making variables represent
statements.) Only such a system as a whole can conceivably contain a
contradiction. Without the formal framework a contradiction doesn't make much
sense.

Here’s a sampling of comments on this topic received from
various mathematicians:

Personally
I think that mathematicians are sometimes too hung up on the details of
formalizations and lose track of the actual math they are talking about. As
the old saying goes, mathematicians are Platonists on weekdays and formalists
on Sunday. I personally think that this tendency is a good and healthy one!

But
Goedel's Theorem can be derived even with a substantially weakened Axiom of
Induction, and I believe that such a weakened axiom would then also lead to a
contradiction, so it seems that we would have to throw away all induction.

If
the consistency of FOL is somehow a hypothetical matter, why should I assume
that you or I make any sense whatever in our babbling?

If
arithmetic was inconsistent, wouldn't all the bridges fall down?

If
so, it's an awfully trivial point, and hardly worth making. She might as
well say "Everything we think might have an error in it because some
demon somewhere is messing with our brains." Quite true. So what?

A
proof from ZF brings with it the supreme confidence that a century of working
with ZF and beyond has given us.

One mathematician argued that (apparently with a straight
face) “The consistency of ZFC is provable in ZFC+Con(ZFC), the consistency of
ZFC+Con(ZFC) is provable in FC+Con(ZFC)+Con(ZFC+Con(ZFC)), etc., so the
infinite hierarchy of such systems provides a complete proof of the consistency
of ZFC. Or, just to press the point, there is actually a system which proves
the consistency of any system... This system even proves its own
consistency.” This proposed hierarchy of systems possesses an interesting
structure. For example, each system has a successor which is assumed to be a
system, and distinct from any of the previous systems (i.e., no loops). And of
course it's necessary to justify speaking of the completed infinite hierarchy,
implying induction. The whole concept might almost be formalized as a set of
axioms along the lines of:

(i) ZFC is a System.

(ii) For any System X, the successor X+con(X) is also a
System.

(iii) ZFC is not the successor of a System.

(iv) If the successors of Systems X and Y are the same,
then X and

Y are the same.

(v) If ZFC is in a set M, and if for every System X in M
the

successor X+con(X) is in M, then M contains every
System.

Now we're getting somewhere! If we can just prove this is
consistent... By the way, here's an interesting quote from Ian Stewart's book
"The Problems of Mathematics":

Mathematical
logicians study the relative consistency of different axiomatic theories in
terms of consistency strength. One theory has greater consistency
strength than another if its consistency implies the consistency of the other
(and, in particular, if it can model the other). The central problem in
mathematical logic is to determine the consistency strength in this ordering
of any given piece of mathematics. One of the weakest systems is ordinary
arithmetic, as formalized axiomatically by Giuseppe Peano... Analysis finds
its place in a stronger theory, called second-order arithmetic. Still
stronger theories arise when we axiomatize set theory itself. The standard
version, Zermelo-Frankel Set Theory, is still quite weak, although the gap
between it and analysis is large, in the sense that many mathematical results
require MORE than ordinary analysis but LESS than the whole of ZF for their
proofs.

It should be noted that Peano's Postulates (P1-P4 in Andrews,
"An Introduction to Mathematical Logic and Type Theory") assert the
existence of an infinitely large set. Since it is not possible to simple ‘exhibit’
an infinite model, any such assertion must simply assume that it is
possible to speak about such things without the possibility of contradictions.
And indeed the Axiom of Infinity is one of the axioms of ZF. Hence the
assertion that PA can be “proved” within ZF can be taken as nothing more than
a joke.

Much of this depends on the status of induction. Normally
we are careful to distinguish between common induction and mathematical
induction. The former consists of drawing general conclusions empirically
from a finite set of specific examples. The latter is understood to be an
exact mathematical technique that, combined with the other axioms of
arithmetic, can be used to rigorously prove things about infinite sets of
integers. For example, by examining the square number 25 we might observe
that it equals the sum of the first five odd integers, i.e., 52 =
1+3+5+7+9. We might then check a few more squares and by common induction
draw the general conclusion that the square of N is always equal to the sum
of the first N odd numbers. In contrast, mathematical induction would
proceed by first noting that the proposition is trivially true for the case
N=1. Moreover, if it's true for any given integer n it is also true for n+1
because (n+1)2- n2
equals 2n+1, which is the (n+1)th odd number. Thus, by mathematical
induction it follows that the proposition is true for all N.

Understandably, many mathematicians take it as an insult
to have mathematical induction confused with common induction. The crucial
difference is that MI requires a formal implicative relation connecting all
possible instances of the proposition, whereas CI leaps to a general
conclusion simply from the fact that the proposition is true for a finite
number of specific instances. Of course, it's easy to construct examples
where CI leads to a wrong conclusion but, significantly, CI often leads to
correct conclusions. We could devote an entire discussion to "the
unreasonable effectiveness of common thought processes", but suffice it
to say that for a system of limited complexity the possibilities can often be
"spanned" by a finite number of instances.

In any case, questions about the consistency of arithmetic
may cause us to view the distinction between MI and CI in a different light.
How do we know that (n+1)2-
n2 always equals 2n+1? Of course this is a trivial example; in
advanced proofs the formal implicative connection can be much less
self-evident. Note that when challenged as to the absolute consistency of
formal arithmetic, one response was to speak of "the supreme confidence
that a century of working with ZF has given us". This, of course, is
nothing but common induction. So too are claims that arithmetic must be
absolutely consistent because otherwise bridges couldn't stand up and check books
wouldn't balance. (These last two are not only common induction, they are bad
common induction.)

Based on these reactions, we may wonder whether,
ultimately, the two kinds of induction really are as distinct as is generally
supposed. It would seem more accurate to say that mathematical induction
reduces a problem to a piece of common induction in which we have the very
highest confidence, because it represents the pure abstracted essence of
predictability, order, and reason that we've been able to infer from our
existential experience. Nevertheless, this inference is ultimately nothing
more (or less) than common induction.

It's clear that many people are highly disdainful of
attempts to examine the fundamental basis of knowledge. In particular, some
mathematicians evidently take it as an affront to the dignity and value of
their profession (not to mention their lives) to have such questions raised.
(One professional mathematician objected to my quoting from Morris Kline's
"Mathematics and the Loss of Certainty", advising me that it is “a
very very very very very very pathetic and ignorant book”.) In general, I
think people have varying thresholds of tolerance for self-doubt. For many
people the exploration of philosophical questions reaches its zenith at the
point of adolescent sophistry, as in "did you ever think that maybe none
of this is real, and some demon is just messing with our minds?" Never
progressing further, for the rest of their lives whenever they encounter an
issue of fundamental doubt they project their own adolescent interpretation
onto the question and dismiss it accordingly.

In any case, this discussion has provided some nice examples
of reactions to such questions, including outrage, condescension, bafflement,
fascination, and complete disinterest. The most controversial point seems to
have been my contention that every formal system is inconsistent. I
was therefore interested to read in Harry Kessler’s “Diaries of a
Cosmopolitan” about a discussion that Kessler had had at a dinner party in Berlin
in 1924.

I
talked for quite awhile to Albert Einstein at a banker's jubilee banquet
where we both felt rather out of place. In reply to my question what problems
he was working on now, he said that he was engaged in thinking. Giving
thought to almost any scientific proposition almost invariably brings
progress with it. For without exception, every scientific proposition was
wrong. That was due to human inadequacy of thought and inability to
comprehend nature, so that every abstract formulation about it was always
inconsistent somewhere. Therefore, every time he checked a scientific
proposition, his previous acceptance of it broke down and led to a new, more
precise formulation. This was again inconsistent in some respects and
consequently resulted in fresh formulations, and so on indefinitely.